# Resolving tiny structures

1. Jan 21, 2016

### Anton Alice

Hello,

I was wondering, if the de-Broglie-relation for particle waves already includes relativistic effects?

Suppose I want to resolve an atomic structure of about, say, 0.1 nanometers. For an attempt using particle waves I would need a certain momentum p=h/0.1nm, at least.
Now comes the question: The particle wave sees the contracted version of the atomic structure. Doesnt this mean that I have to include a gamma into the nominator? p would then be gamma*m*v, and then the gammas would cancel, leading to: mv=h/0.1nm, whereby mv stands for the classical momentum. Something is wrong here...

EDIT:
I think my problem is, that I mixed up different reference frames. I have to stay in one reference frame, which I chose to be the the one of the observer. The observer observes an atomic structure of 0.1nm, and observes a particle with speed v (and corresponding p). These two observations have to fit together, i.e. the de-Broglie-relation needs to be fulfilled. And if it is fulfilled in one reference frame, then also in any other. Like saying, if barack obama has been elected in one reference frame, then also he is or will be in any other.
Can you elaborate on this?

Last edited: Jan 21, 2016
2. Jan 21, 2016

### Staff: Mentor

That's pretty much it.

3. Jan 21, 2016

### Staff: Mentor

Are you familiar with four-vectors? The relativistic de Broglie relationship is simply $p=\hbar k$ where p is the four momentum and k is the four-wavevector.